Question

Calcule a derivada.

y=(2x^4)/(b² - x²)

Answer 1

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Calcular a derivada da função:

$\mathsf{y=\dfrac{2x^4}{b^2-x^2}\qquad\qquad (com~b~constante)}$


Para a derivada desta função, usamos a regra do quociente:

$\mathsf{\dfrac{dy}{dx}=\dfrac{d}{dx}\!\left(\dfrac{2x^4}{b^2-x^2}\right)}\\\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{\frac{d}{dx}(2x^4)\cdot (b^2-x^2)-2x^4\cdot \frac{d}{dx}(b^2-x^2)}{(b^2-x^2)^2}}\\\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{4\cdot 2x^{4-1}\cdot (b^2-x^2)-2x^4\cdot (0-2x^{2-1})}{(b^2-x^2)^2}}\\\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{8x^3\cdot (b^2-x^2)-2x^4\cdot (-2x^1)}{(b^2-x^2)^2}}\\\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{8x^3\cdot (b^2-x^2)+2x^4\cdot 2x}{(b^2-x^2)^2}}$

$\mathsf{\dfrac{dy}{dx}=\dfrac{8x^3\cdot b^2-8x^3\cdot x^2+4x^5}{(b^2-x^2)^2}}\\\\\\ \mathsf{\dfrac{dy}{dx}=\dfrac{8b^2x^3-8x^5+4x^5}{(b^2-x^2)^2}}$


∴      $\boxed{\begin{array}{c} \mathsf{\dfrac{dy}{dx}=\dfrac{8b^2x^3-4x^5}{(b^2-x^2)^2}} \end{array}}$          ✔


Bons estudos! :-)


Tags:  derivada função regra do quociente cálculo diferencial integral